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16 Sep 2014, 01:30
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If the least common multiple of a positive integer \(x\), \(4^3\) and \(6^5\) is \(6^6\), then how many values can \(x\) take?
A. \(1\)
B. \(6\)
C. \(7\)
D. \(30\)
E. \(36\)
A. \(1\)
B. \(6\)
C. \(7\)
D. \(30\)
E. \(36\)
16 Sep 2014, 01:30
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Official Solution:
If the least common multiple of a positive integer \(x\), \(4^3\) and \(6^5\) is \(6^6\), then how many values can \(x\) take?
A. \(1\)
B. \(6\)
C. \(7\)
D. \(30\)
E. \(36\)
We are given that \(6^6=2^{6}*3^{6}\) is the least common multiple (LCM) of three numbers:
\(x\);
\(4^3=2^6\);
\(6^5 = 2^{5}*3^5\);
First notice that \(x\) cannot include any prime factors other than 2 and/or 3 as the LCM is exclusively composed of these primes.
Next, since the exponent of 3 in the LCM is greater than the exponent of 3 in the other two numbers, \(x\) must have \(3^6\) as a factor. Otherwise, the factor \(3^{6}\) wouldn't appear in the LCM.
Furthermore, \(x\) can include the prime number 2 raised to any power from 0 to 6, inclusive (the LCM limits the power of 2 in \(x\) to 6, so it cannot be any higher).
Therefore, \(x\) can take on a total of 7 distinct values, namely:
\(3^6\);
\(2*3^6\);
\(2^2*3^6\);
\(2^3*3^6\);
\(2^4*3^6\);
\(2^5*3^6\);
\(2^6*3^6\).
Answer: C
If the least common multiple of a positive integer \(x\), \(4^3\) and \(6^5\) is \(6^6\), then how many values can \(x\) take?
A. \(1\)
B. \(6\)
C. \(7\)
D. \(30\)
E. \(36\)
We are given that \(6^6=2^{6}*3^{6}\) is the least common multiple (LCM) of three numbers:
\(x\);
\(4^3=2^6\);
\(6^5 = 2^{5}*3^5\);
First notice that \(x\) cannot include any prime factors other than 2 and/or 3 as the LCM is exclusively composed of these primes.
Next, since the exponent of 3 in the LCM is greater than the exponent of 3 in the other two numbers, \(x\) must have \(3^6\) as a factor. Otherwise, the factor \(3^{6}\) wouldn't appear in the LCM.
Furthermore, \(x\) can include the prime number 2 raised to any power from 0 to 6, inclusive (the LCM limits the power of 2 in \(x\) to 6, so it cannot be any higher).
Therefore, \(x\) can take on a total of 7 distinct values, namely:
\(3^6\);
\(2*3^6\);
\(2^2*3^6\);
\(2^3*3^6\);
\(2^4*3^6\);
\(2^5*3^6\);
\(2^6*3^6\).
Answer: C
akhilvarma8
GMAT 2: 660 Q45 V35
Posts: 19
21 Jun 2019, 10:44
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Hi,
It took me a lot of time to understand this answer explanation but after I re-read my notes about LCM I was able to understand it to the full.
To all others who are still struggling, the rule of LCM is "When you combine two factor trees (prime factors) of two numbers to arrive at an LCM that contain overlapping primes, you drop the overlap".
In this case, it is given that 6^6 = 2^6 x 3^6 is the LCM of
X as "A"
4^3 = 2^6 as "B"
6^5 = 2^5 and 3^5 as "C"
Now, the LCM will have 2^6 from B, dropping out 2^5 from C, 3^6 from A (thinking we start with 3^6), dropping out 3^5 from C.
Now, as we ware asked to give all the possible values of X, we can write down that it can have
3^6
2 x 3^6
2^2 x 3^6
2^3 x 3^6
2^4 x 3^6
2^5 x 3^6
2^6 x 3^6
as all the values with 2 power will be dropped for overlapping with other numbers. Note that 2^7 x 3^6 will be wrong as it will increase the number of 2s adding one more to the LCM which is not what was given in the question.
This is my first ever explanation to an answer on GC. Hope you found it helpful. If not sorry for the confusion.
Thank you!!
It took me a lot of time to understand this answer explanation but after I re-read my notes about LCM I was able to understand it to the full.
To all others who are still struggling, the rule of LCM is "When you combine two factor trees (prime factors) of two numbers to arrive at an LCM that contain overlapping primes, you drop the overlap".
In this case, it is given that 6^6 = 2^6 x 3^6 is the LCM of
X as "A"
4^3 = 2^6 as "B"
6^5 = 2^5 and 3^5 as "C"
Now, the LCM will have 2^6 from B, dropping out 2^5 from C, 3^6 from A (thinking we start with 3^6), dropping out 3^5 from C.
Now, as we ware asked to give all the possible values of X, we can write down that it can have
3^6
2 x 3^6
2^2 x 3^6
2^3 x 3^6
2^4 x 3^6
2^5 x 3^6
2^6 x 3^6
as all the values with 2 power will be dropped for overlapping with other numbers. Note that 2^7 x 3^6 will be wrong as it will increase the number of 2s adding one more to the LCM which is not what was given in the question.
This is my first ever explanation to an answer on GC. Hope you found it helpful. If not sorry for the confusion.
Thank you!!
